The wave function of a standing wave is y(x, t) = 4.44 mm sin[(32.5 rad/m)x] sin[(754 rad/s)t]. For the two traveling waves that make up this standing wave, find the (c) frequency.

Verified step by step guidance

1. Understand that a standing wave is formed by the superposition of two traveling waves moving in opposite directions. The given wave function is a product of two sine functions, one depending on position x and the other on time t.
2. Identify the angular frequency \( \omega \) from the time-dependent part of the wave function, which is \( \sin[(754 \text{ rad/s})t] \). Here, \( \omega = 754 \text{ rad/s} \).
3. Recall the relationship between angular frequency \( \omega \) and frequency \( f \), given by \( \omega = 2\pi f \).
4. Solve for the frequency \( f \) using the formula \( f = \frac{\omega}{2\pi} \). Substitute \( \omega = 754 \text{ rad/s} \) into the equation.
5. Calculate the frequency \( f \) by dividing \( 754 \text{ rad/s} \) by \( 2\pi \). This will give you the frequency of the traveling waves that form the standing wave.

Key Concepts

Standing Waves

Standing waves are formed by the superposition of two traveling waves moving in opposite directions with the same frequency and amplitude. They are characterized by nodes, where the wave amplitude is always zero, and antinodes, where the amplitude is maximum. Understanding standing waves is crucial for analyzing the given wave function.

Wave Function

A wave function describes the displacement of a wave at any point in space and time. In the given function y(x, t) = 4.44 mm sin[(32.5 rad/m)x] sin[(754 rad/s)t], the spatial part (32.5 rad/m) relates to the wave number, and the temporal part (754 rad/s) relates to the angular frequency. This function helps in identifying the properties of the wave, such as frequency and wavelength.

Frequency and Angular Frequency

Frequency (f) is the number of oscillations per unit time, measured in hertz (Hz), while angular frequency (ω) is the rate of change of the wave's phase, measured in radians per second. They are related by the formula ω = 2πf. In the given wave function, the angular frequency is 754 rad/s, allowing us to calculate the frequency as f = ω/2π.